Publish Wizard

In fact you can publish your solution without using the Publish Wizard so this term can be a little misleading.
The Publish Wizard deployment method is an alternative way of deploying a solution rather than using a Setup Project which uses Windows Installer.
There are actually 2 ways you can quickly publish your solutions.
1) Properties > Publish Tab
2) Build > Publish Wizard

The Publish Wizard creates a deployment manifest for the solution and copies all of the required files to the deployment location.

The Publish Wizard does not grant full trust to the add-in assembly for the end user
When you deploy using the Publish Wizard you must also deploy the required security policy to each end user.

Build > Publish Solution

This can alternatively be run by right clicking on the solution and selecting Publish.
By default this will default to the Publish Location on the Publish tab.
Click finish to confirm the location and publich.
Confirmation is also shown in the status bar
The details can also be viewed in the Output window
In the "Show Output from" drop-down select Build to see the build output details
SS of wizard

Publish Now

When you press Publish Now, the following things happen:
1) A vsto deployment manifest (.vsto) is created
2) Copy the manifest and update the application manifest
however it wont
3) Create any necessary registry entries which are required for application level add-ins
4) Doesn't grant full trust to your assemblies in the security ploicy of the end user.
5) Check or install any pre-requisitis (like the PIAs or VSTO runtime)

You can use the Publish Wizard to publish VSTO solutions to a network location or http.
This creates both the application and deployment manifests automatically and copies all the files to the deployment location.

Every time you publish your solution, VSTO creates a new folder containing your entire solution (copy of the solution file and the deployment manifest)
This separation allows you to easily rollback or forward to a different version
You will need to remove old versions manually from this folder as you see fit.

Using the Publish Wizard uses XML manifest files identical to ClickOnce.
This allows you to easily deploy an application to a web server or network.
Allows applications to be run on a URL. When the user clicks on the URL, the runtime automatically downloads the application, and all related assemblies, to a special download cache.

Wont automatically check that the client has the necessary pre-requisites
Wont automatically create the necessary permissions